# Arithmetic and Geometric Mean Returns

There is more than one way of both assessing mean returns and assessing risk and we are going to look at two definitions on each side. Let’s go back to our data set.

Year |
USA |
Spain |
Egypt |
World |

2004 |
10.7% | 29.6% | 126.2% | 15.8% |

2005 |
5.7% | 4.9% | 161.6% | 11.4% |

2006 |
15.3% | 50.2% | 17.1% | 21.5% |

2007 |
6.0% | 24.7% | 58.4% | 12.2% |

2008 |
-37.1% | -40.1% | -52.4% | -41.8% |

2009 |
27.1% | 45.1% | 39.7% | 35.4% |

2010 |
15.4% | -21.1% | 12.4% | 13.2% |

2011 |
2.0% | -11.2% | -46.9% | -6.9% |

2012 |
16.1% | 4.7% | 47.1% | 16.8% |

2013 |
32.6% | 32.3% | 8.2% | 23.4% |

AM |
9.4% |
11.9% |
37.2% |
10.1% |

GM |
7.6% |
7.9% |
21.4% |
7.7% |

**The definition of the arithmetic mean return**

To compute the arithmetic mean return you basically add up all the observations and divide that sum of observations by the number of observations. So you do that for the US, you add up those ten annual returns and you divide by ten or you do the same thing for Spain or for Egypt or for the world market, those are the numbers that you are going to get. 9.5 for the US, 11.9 for Spain, 37.2 for Egypt, and 10.1 for the world market.

*What is the meaning of these numbers?*

Well, the meaning of these numbers, and here is one first important idea. For you to keep in mind, is that this arithmetic mean return is something that we do not use a whole lot in finance. Most of the time when we talked about, the mean return of an asset we mean something else. But this arithmetic mean return more often than not, is actually not what we mean. I’m not saying that this is a useless number. I’m saying that when we describe the performance of an asset, this is not the relevant number that we tend to use.

*What does that 9.4 percent mean?*

Well 9.4%, first of all, it has the same interpretation as any average. There have been high returns, low returns, positive returns, negative returns when you average all those you get 9.4. So basically, you know, that is a number that maybe its usefulness is to compare it to the number for Spain, to compare it to the number of Egypt, to compare to the number of the world market, or any other asset that you may want to compare it to and see which one is higher, which one is lower. Of course it will be an incomplete comparison, because we need to bring risk into the equation, but one possibility, you know, in the same way, we could actually be measuring the age, of the, all the people taking these smoke, and compare it to the age of the people taking some other mock. And to make a distinction, you know, where are the younger people or where are the older people, or where are the taller people or where are the skinnier people. We could make all those comparisons, and, you know the usefulness of that comparison will depend on how interesting is the question. But that you actually post, but that arithmetic mean return is nothing but the average of everything that happened. **High returns, low return, positive returns, negative returns**, average all that, and you get, in the case of the US for example 9.4. percent. Now 9.4% some people were also referred to it as the return in the typical period. And again this goes back to the same idea. There have been returns that have been high or low, positive or negative in the typical average period you get a 9.4. Percent of return and that is as far as this measure, of mean return goes there’s not much more to say. So if we go back to the two definitions, just to actually have them there. It is again we look back, we take the average of the returns and that is simply what it is, and number two. Given that the returns have been high, low, positive, or negative in the typical period they’ve been at that particular number which we refer to, as the arithmetic mean return.

**Definition of Geometric mean return**

The second definition is what we call **the geometric mean return**. This looks and sounds a little bit more difficult and the calculation is, in fact, a little bit more difficult than the arithmetic mean return. Let’s start with the numbers. If we go back to our data, and we actually calculate ex, using the formula that you’re going to see. And the expression that you are going to see the technical note. Then you will get those numbers. 7.6% for the US, 7.9% for Spain, 21.4 for Egypt, and 7.7%. For the world market. And the first thing you see there when you compare, country by country, the arithmetic and the geometric mean return, is that these two numbers are different. In fact, not only they are different, but also you should be able to see one more thing and that is that the arithmetic being returned in all cases is higher than the geometric being returned. Now, let’s think a little bit and try to interpret that geometric being returned. And this part is really important. It is important because whenever in finance we talk about the mean annual return of an asset, then basically this is the number that we’re referring to. So let’s think a little bit about the world market. Therefore, that is the world market and we are repeating all those numbers here as well as the geometric mean return of 7.7%. All right, now think; let us think about, this in the following way. Let’s suppose that you take $100 out of your pocket at the very beginning of the year 2004. And as you see in that table, if during the year 2004, the return that you got was 15.8%. So that basically means that if you actually take $100 out of your pocket at the beginning of 2004, or what is exactly the same, at the very end of 2003, and you get a 15.8% return, then at the end of the year 2004. You ended up with $115 and, $8 in your, in your pocket. Now, if you actually keep that investment, and you go through the year 2005, invested in the same world market and you get an 11.4% return. That means that you started with 115.8, you obtained an 11.4%, beginning from that amount and at the end of the period you have 128.9. If you keep doing that over and over again, when you get to the beginning of the year 2013, you have in your pocket 170.5 dollars. And in the year 2013, you obtain a 23.4% return, which means that at the end of these ten years being invested, you get, the number that you see on the screen, which is $210.4, $210.40. That means that if you had taken a hundred dollars out of your pocket at the very end of 2003, and had remained invested in the world market. For the subsequent ten years, at the end of those ten years, you would have in your pocket $210.40. And here comes the interesting part. And that is, let’s think about it in a slightly different way. Let’s suppose you take $100 out of your pocket. And you actually get a 7.7 mean annual return, over and over and over again. So, in other words, imagine, just imagine that you get 7.7 on top of 7.7 on top of 7.7, ten times in a row. And that you let the capital accumulate over time.

Guess what you would get, at the end of that?

You would get exactly, the same $210.40. So if you compare those two numbers, then you get what the interpretation, of the geometric mean return, is. If you have been invested in the world market. Between the years 2004 and 2013, starting with $100, you would have ended with $210.40. If you had obtained a mean annual compound return of 7.7% over those ten years, you would have ended up. With exactly the same amount of money. So that gives you the interpretation of the **geometric mean is the mean annual compound return that you actually obtain by investing **in the world market at the end of 2003, and de-investing, or getting out of the world market at the end of, 2013. And that is a, a very important way, and a very important definition of talking about, mean. Mean returns. So, two things to keep in mind. First, that the geometric mean return is the average rate at which an invested capital evolved over time. Of course, you were getting different returns over time, year after year you got positive returns, negative returns, high returns, and low returns. But if you actually think in terms of the mean annual evolution of your capital, in the case of the world market, it evolved at 7.7%, so it’s the average rate, in our case annual because we’re using annual data. Is the average annual rate of which a capital invested, evolved over time? It basically gives you, what you gain or lose, in any given period, compounded. And that keyword compounded, is very important because it basically tells you. At the, now it is the idea of saying over and over and over again. That 7.7% that we looked at before. If you had gotten that over and over and over and over, over ten years, then you would have gotten to exactly the same amount of money as being exposed to all those returns that we’ve seen. In the ten years that we explore. So, when, whenever you hear someone talking about the mean annual compound return of an asset, they’re basically referring to that geometric mean return.